Answer each of the following:

a. An internal revenue service auditor knows that \(3 \%\) of all income tax forms contain errors. Returns are assigned randomly to auditors for review. What is the probability that an auditor will have to view four tax returns until the first error is observed? That is, what is the probability of observing three returns with no errors, and then observing an error in the fourth return?

b. Let \(Y\) be the number of independent trials of an experiment before a success is observed. That is, it is the number of failures before the first success. Assume each trial has a probability of success of \(p\) and a probability of failure of \(1-p\). Is this a discrete or continuous random variable? What is the set of possible values that \(Y\) can take? Can \(Y\) take the value zero? Can \(Y\) take the value 500?

c. Consider the \(p d f f(y)=P(Y=y)=p(1-p)^{y}\). Using this \(p d f\) compute the probability in (a). Argue that this probability function generally holds for the experiment described in (b).

d. Using the value \(p=0.5\), plot the \(p d f\) in (c) for \(y=0,1,2,3,4\).

e. Show that \(\sum_{y=0}^{\infty} f(y)=\sum_{y=0}^{\infty} p(1-p)^{y}=1\). If \(|r|<1\) then \(1+r+r^{2}+r^{3}+\cdots=1 /(1-r)\).]

f. Verify for \(y=0,1,2,3,4\) that the \(c d f P(Y \leq y)=1-(1-p)^{y+1}\) yields the correct values.